Non-randomness that arises in the shuffling of multiple numbers of decks of playing cards is modeled. An efficient way to exploit the non-randomness is derived, and its effect on the win rate for a simple game of chance is derived.

Random processes of the diffusion type have the property that microscopic fluctuations of the trajectory make possible the identification of certain statistical parameters from one continuous observation. The paper deals with the construction of parameter estimates when observations are made at discrete but very dense time points.

A correlated random walk is studied in which, at each stage, the velocity changes according to a first-order process. Motion is considered both with and without friction, the former situation being the discrete analogy of the Uhlenbeck–Ornstein process. Exact and limiting expressions are developed for the cumulant structures.

We study monotone and associated Markov chains on finite partially ordered state spaces. Both discrete and continuous time, and both time-homogeneous and time-inhomogeneous chains are considered. The results are applied to binary and multistate reliability theory.

A general method to obtain insensitive upper and lower bounds for the stationary distribution of queueing networks is sketched. It is applied to an overflow model. The bounds are shown to be valid for service distributions with decreasing failure rate. A characterization of phase-type distributions with decreasing failure rate is given. An approximation method is proposed. The methods are illustrated with numerical results.

In this paper various notions of positive and negative dependence for bivariate stochastic processes are introduced and their interrelationship is studied. Examples are given to illustrate these concepts.

Cowan and Zabczyk (1978) have studied a continuous-time generalization of the so-called secretary problem, where options arise according to a homogeneous Poisson processes of known intensity λ. They gave the complete strategy maximizing the probability of accepting the best option under the usual no-recall condition. In this paper, the solution is extended to the case where the intensity λ is unknown, and also to the case of an inhomogeneous Poisson process with intensity function λ (t), which is either supposed to be known or known up to a multiplicative constant.

Many mean-value quantities of stationary random tessellations can be expressed in terms of three fundamental mean-value quantities. In this note we characterize the set of triples of mean values that can be realized, and show that every possible triple can arise from a suitable ergodic stationary isotropic tessellation.

We provide sufficient conditions under which two random vectors could be stochastically compared using the standard construction. These conditions are weaker than those discussed by Arjas and Lehtonen (1978) and Veinott (1965). Using these conditions we present extensions of (i) a result of Block et al. (1984) concerning the stochastic monotonicity of independent and identically distributed random variables conditioned on their partial order statistics, and (ii) a theorem of Efron (1965) regarding an increasing property of Pólya frequency functions. Applications of these extensions are also pointed out.

We consider general queueing models dealing with multiple classes of customers and address the question under what conditions and in what (stochastic) sense the marginal increase in various performance measures, resulting from the addition of a new class of customers to an existing system, is larger than if the same class were added to a system dealing with only a subset of its current customer base.

Our results enhance our understanding of the dependence of various performance measures with respect to the composition of the customer base. In addition they translate readily into convexity results in an (appropriately defined) arrival rate.

Arcs of length lk, 0 < lk < 1, k = 1, 2, ···, n, are thrown independently and uniformly on a circumference having unit length. Let P(l1, l2, · ··, ln) be the probability that is completely covered by the n random arcs. We show that P(l1, l2,· ··, ln) is a Schur-convex function and that it is convex in each argument when the others are held fixed.

The theory of robust non-linear filtering in Clark (1978) and Davis (1980), (1982) is used to evaluate the limiting conditional distribution of a diffusion, given an observation of a ‘rare-event' sample-path of the diffusion, as the signal-to-noise ratio and the diffusion noise-intensity converge to infinity and zero respectively. Under mild conditions it is shown that the limiting conditional distribution is a Dirac measure concentrated at a trajectory which solves a variational problem parametrised by the sample-path of the observed signal.

This paper examines a general performance measure for queueing systems. This criterion reflects both the mean and the variance of sojourn times; the standard deviation is a special case. The measure plays a key role in certain production models, and it should be useful in a variety of other applications.

We focus here on convexity properties of an approximation of the measure for the M/G/c queue. For c ≧ 2 we show that this quantity is convex in the arrival rate. Assuming the service rate acts as a scale factor in the service-time distribution, the measure is convex in the service rate also.

This paper considers single-server, multi-queue systems with cyclic service. Non-zero switch-over times of the server between consecutive queues are assumed. A stochastic decomposition for the amount of work in such systems is obtained. This decomposition allows a short derivation of a ‘pseudo-conservation law' for a weighted sum of the mean waiting times at the various queues. Thus several recently proved conservation laws are generalised and explained.

A composite material is a parallel arrangement of stiff brittle fibers in a flexible matrix. Under load fibers fail, and the loads of failed fibers are locally redistributed onto nearby survivors through the matrix. In this paper we develop a new technique for computing the probability of failure under a previously studied model of the failure process. A recursion and limit theorem are obtained which apply separately to static strength and fatigue lifetime depending on the composite loading and the probability model for the failure of individual fibers under their own loads. The limit theorem yields an approximation for the distribution function for composite lifetime which is of the form 1 – [1 – W(t)]mn where W(t) is a characteristic distribution function and mn is the composite volume, reflecting a size effect. A similar result holds also for static strength. In both cases such a result was conjectured several years ago. This limit theorem is obtained from the recursion upon applying a key theorem in the theory of the renewal equation. In the proofs three technical conditions arise which must be verified in specific applications. In the case of static strength these conditions are quite easy to verify, but in the case of fatigue lifetime the verification is generally difficult, and entails considerable numerical computation.

n jobs are to be preemptively scheduled for processing on n machines. The machines may have differing speeds and the jobs have processing requirements which are distributed as independent exponential random variables with different means. Holding cost g(U) is incurred per unit time that the set of uncompleted jobs is U and it is desired to minimize the total expected holding cost which is incurred until all jobs are complete. We show that if g satisfies certain simple conditions then the optimal policy is one which takes the jobs in the order 1, 2, ···, n and assigns each uncompleted job in turn to the fastest available machine. In the special case in which the objective is to minimize the expected weighted flowtime, where there is a holding cost of wi while job i is incomplete, the sufficient condition is simply w1 ≧ … ≧ wn and λ1 w1 ≧ … ≧ λn wn.

There are a number of ad hoc regression models for the statistical analysis of lifetime data, but only a few examples exist in which physical considerations are used to characterize the model. In the present paper a complete characterization of a regression model is given by solving a functional equation recurring in the literature for the case of a fatigue problem. The result is that, if the lifetime for given values of the regressor variable and the regressor variable for a given lifetime are both Weibull variables (assumptions which are well founded, at least as approximations, from extreme-value theory in some concrete applications), there are only three families of (conditional) distribution for the lifetime (or for the regressor variable). This model is then applied to a practical problem for illustration.

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.

In this paper, the extremal properties of the ergodic sojourn times in G/G/1queues under various service disciplines are studied in terms of the convex ordering. It is shown that among work-conserving non-preemptive service disciplines that are service time independent, the FIFO and the LIFO service disciplines provide the minima and the maxima, respectively, of the ergodic sojourn times for any G/G/1 queue. For G/M/1 queues, this class of work-conserving service disciplines is extended to include preemptive/resume disciplines. In this case the FIFO and LIFO-P (preemptive/resume LIFO) service disciplines attain the minima and maxima, respectively. These extend results of Durr (1971), Kingman (1962) and a recent result of Ramaswami (1984). Further results are obtained for G/Em/1 and G/D/1 queues.

A general shock model in which the time intervals between shocks have infinite expectation is considered. Limit theorems for the first time the magnitude of a shock exceeds a and the historical maximum magnitude are given.